double pendula lead to chaotic dynamics: good. but what happens if we make things a bit more… complicated? what about a triple pendulum? would it be even more chaotic? let’s see…

well, why stop there? if we take this to a ridiculous extreme, we get ridiculously complicated behavior…

of course, there’s no reason to consider only chains of pendula. the following compound pendulum has multiple bars and ends, and it is one of my favorite things to look at.

of course, instead of shaking a pendulum horizontally or vertically, one can go meta and hang a pendulum from a pendulum. let’s see what happens with a such a double pendulum.

that looks chaotic, doesn’t it? but what is meant by chaos? that takes a long time to learn properly. for now, let’s focus on one crucial requirement for chaos: sensitive dependence on initial conditions. this is the idea that nearby initial conditions diverge in behavior exponentially quickly. let’s take a look:

why is the double pendulum chaotic when the single pendulum is not (unless shaken in the right way)? it’s all about topology. the configuration space of a single pendulum is two-dimensional — one for the angle and one for the angular velocity. the configuration space is the tangent bundle of a circle, homeomorphic to an annulus. that’s not enough room for a flow to get chaotic. but the double pendulum has twice the dimension — it’s a 4-d tangent bundle of a torus — and that is plenty of room for chaos to creep in.

several commenters on twitter/youtube liked the horizontally shaken pendula and asked about what was being represented. there were many pendula with different initial positions — all initial velocities were set to zero. there were several queries as to what would happen with different lengths or different masses… would resonances with the vibration and the natural frequencies kick in?

dynamical systems studies *interesting* behaviors in systems. one such class of interesting behaviors is equilibria, with stable and unstable varieties. but there’s more. consider a slight variation on the shaken inverted pendulum, where the shaking is happening horizontally. this leads to interesting and varied behaviors, depending (somewhat sensitively) on the initial conditions. some are *periodic* (they repeat over time); others seem to keep evolving and keep getting more & more complex. what is happening? how do we make sense of such? oh, there’s much more to learn about dynamical systems…

dynamical systems begins with the notion of equilibria — states that remain constant over time. In the simplest cases, one distinguishes between stable and unstable equilibria, though to specify what exactly these mean requires care (and math).

one simple example of stable and unstable equilibria is found in a rigid-rod pendulum, rotating about a pivot point (with or, as here, without, friction). letting the system evolve from a variety of initial conditions reveals a stable equilibrium (bottom) and an unstable equilibrium (top).

there’s a lot to discuss about this system. skipping ahead (for the sake of seeing something wonderful), let’s consider what happens when you give this system a shake, vibrating the pivot point up & down. the simulation indicates that past a criticla threshold of amplitude/frequency, shaking the pendulum converts the unstable equilibrium to a stable equilibrium.

that’s swell. but does the mathematics prove that this is the expected behavior? well, that’s one of the reasons why you take a dynamical systems course.

in anticipation of teaching introductory dynamical systems & bifurcations to engineering students at penn this fall, i’m going to chat off and on about visualizing and (more importantly) simulating dynamical systems. this has challenged me in terms of my programming skills, but it’s a lot of fun to see theorems work in simulated systems.

one of the most vexing things for me to illustrate is a planar vector field, especially when there is a singularity. there are so many distractions that arise from the canonical square-grid sampling that is endemic to software. here is an example with the linear vector field dx/dt = x ; dy/dt = -y

now for something completely different: the loxodromic grid, obtained by intersecting two circle’s worth of counterspinning spirals. you see these in sunflowers, pinecones, and the like. it seems as though nobody uses the term loxodromic grid (why was it in my head when i wanted to call this something? why did i know to use 137.5 degrees? intuition is a frustrating thing…). it also seems as though nobody ever plots vector fields on such a grid. pity, since that meant it took me a long time to figure out how to do it. but: it is worth the trouble…

i really like the fact that the sampling gets increasingly tight near the equilibrium. i am simultaneously disturbed and pleased that the stable and unstable manifolds are “hidden” (as it were). it is fitting that you have a hard time visually getting on and staying on the stable manifold, since that’s exactly what happens in the flow. even the visual artefacts are instructive: note that near the origin, the tiny vectors are packed in such a way as to suggest not a circle (where the sampling points are) but a squashed ellipse. this is right: the flow squeezes and stretches areas.

this is not a perfect plot. but it is satisfying to me, after having tried and failed to make square and hex grids not look awful.

it’s been awhile since i’ve maintained this site or posted new content: the consequence of having no followers to disappoint! now that volume 4 of calculus BLUE (the hardest one by far) is done, i can go back to musing about illustration some more.

is there any nice way to generate a mobius strip (from the computer graphics point of view)? i am doubtful. you won’t get it parametrically (normals are messed up; closure is a pain) and don’t try realizing it implicitly in 3-d ðŸ˜‰

i think it helps to have a fattened boundary, as well as to have the surface partially translucent (fresnel shader) with some subtle reflectivity. this definitely looks much better when animated, but you’ll have to wait for BLUE vol 4 to hit youtube for that…

this is day 3 of trying and failing to properly animate stokes’ theorem in 3-d. stills are not hard: it’s the animation that’s rough.

so, as a way to procrastinate (& practice), here is a cartoon drawing of a sheaf of vector spaces over a graph, thought of as an algebraic data structure.

this has a nice greyscale scheme, with good sense of scope. it does not do a very good job of suggesting the heterogeneity of the stalks, nor the linear transformations between them. but you can’t have everything.